HOW SMOOTH IS A DEVIL'S STAIRCASE?

Abstract

Let the self-similar set C in R be defined by [Formula: see text] with a disjoint union, where the hj's are similitude mappings with ratios 0 < aj < 1. Let μ on C be the self-similar probability measure corresponding to the probability vector [Formula: see text], where ξ = dim HC is the Hausdorff dimension of C. Let S be the set of points at which the probability distribution function F of μ has no derivative, finite or infinite. We prove that dim H S = ( dim HC)2 and dim P S = dim B S = dim HC.